Optimal. Leaf size=120 \[ -\frac {2 d^2 \sqrt {c+d x}}{b^3 \sqrt {a+b x}}-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {2 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {49, 65, 223,
212} \begin {gather*} \frac {2 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2}}-\frac {2 d^2 \sqrt {c+d x}}{b^3 \sqrt {a+b x}}-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 65
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx &=-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {d \int \frac {(c+d x)^{3/2}}{(a+b x)^{5/2}} \, dx}{b}\\ &=-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {d^2 \int \frac {\sqrt {c+d x}}{(a+b x)^{3/2}} \, dx}{b^2}\\ &=-\frac {2 d^2 \sqrt {c+d x}}{b^3 \sqrt {a+b x}}-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {d^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{b^3}\\ &=-\frac {2 d^2 \sqrt {c+d x}}{b^3 \sqrt {a+b x}}-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {\left (2 d^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^4}\\ &=-\frac {2 d^2 \sqrt {c+d x}}{b^3 \sqrt {a+b x}}-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {\left (2 d^3\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b^4}\\ &=-\frac {2 d^2 \sqrt {c+d x}}{b^3 \sqrt {a+b x}}-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {2 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 111, normalized size = 0.92 \begin {gather*} -\frac {2 \sqrt {c+d x} \left (15 a^2 d^2+5 a b d (c+7 d x)+b^2 \left (3 c^2+11 c d x+23 d^2 x^2\right )\right )}{15 b^3 (a+b x)^{5/2}}+\frac {2 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{\frac {5}{2}}}{\left (b x +a \right )^{\frac {7}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 219 vs.
\(2 (92) = 184\).
time = 1.93, size = 463, normalized size = 3.86 \begin {gather*} \left [\frac {15 \, {\left (b^{3} d^{2} x^{3} + 3 \, a b^{2} d^{2} x^{2} + 3 \, a^{2} b d^{2} x + a^{3} d^{2}\right )} \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (23 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} + {\left (11 \, b^{2} c d + 35 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}, -\frac {15 \, {\left (b^{3} d^{2} x^{3} + 3 \, a b^{2} d^{2} x^{2} + 3 \, a^{2} b d^{2} x + a^{3} d^{2}\right )} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + 2 \, {\left (23 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} + {\left (11 \, b^{2} c d + 35 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1025 vs.
\(2 (92) = 184\).
time = 1.95, size = 1025, normalized size = 8.54 \begin {gather*} -\frac {\sqrt {b d} d^{2} {\left | b \right |} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b^{5}} - \frac {4 \, {\left (23 \, \sqrt {b d} b^{9} c^{5} d^{2} {\left | b \right |} - 115 \, \sqrt {b d} a b^{8} c^{4} d^{3} {\left | b \right |} + 230 \, \sqrt {b d} a^{2} b^{7} c^{3} d^{4} {\left | b \right |} - 230 \, \sqrt {b d} a^{3} b^{6} c^{2} d^{5} {\left | b \right |} + 115 \, \sqrt {b d} a^{4} b^{5} c d^{6} {\left | b \right |} - 23 \, \sqrt {b d} a^{5} b^{4} d^{7} {\left | b \right |} - 70 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{7} c^{4} d^{2} {\left | b \right |} + 280 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{6} c^{3} d^{3} {\left | b \right |} - 420 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{5} c^{2} d^{4} {\left | b \right |} + 280 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{4} c d^{5} {\left | b \right |} - 70 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{3} d^{6} {\left | b \right |} + 140 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{5} c^{3} d^{2} {\left | b \right |} - 420 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{4} c^{2} d^{3} {\left | b \right |} + 420 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{3} c d^{4} {\left | b \right |} - 140 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{2} d^{5} {\left | b \right |} - 90 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{3} c^{2} d^{2} {\left | b \right |} + 180 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b^{2} c d^{3} {\left | b \right |} - 90 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{2} b d^{4} {\left | b \right |} + 45 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} b c d^{2} {\left | b \right |} - 45 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} a d^{3} {\left | b \right |}\right )}}{15 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{5} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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